1//===-- Implementation header for asin --------------------------*- C++ -*-===//
2//
3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4// See https://llvm.org/LICENSE.txt for license information.
5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6//
7//===----------------------------------------------------------------------===//
8
9#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_ASIN_H
10#define LLVM_LIBC_SRC___SUPPORT_MATH_ASIN_H
11
12#include "asin_utils.h"
13#include "src/__support/FPUtil/FEnvImpl.h"
14#include "src/__support/FPUtil/FPBits.h"
15#include "src/__support/FPUtil/double_double.h"
16#include "src/__support/FPUtil/dyadic_float.h"
17#include "src/__support/FPUtil/multiply_add.h"
18#include "src/__support/FPUtil/sqrt.h"
19#include "src/__support/macros/config.h"
20#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
21#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
22#include "src/__support/math/asin_utils.h"
23
24namespace LIBC_NAMESPACE_DECL {
25
26namespace math {
27
28LIBC_INLINE double asin(double x) {
29 using namespace asin_internal;
30 using FPBits = fputil::FPBits<double>;
31
32 FPBits xbits(x);
33 int x_exp = xbits.get_biased_exponent();
34
35 // |x| < 0.5.
36 if (x_exp < FPBits::EXP_BIAS - 1) {
37 // |x| < 2^-26.
38 if (LIBC_UNLIKELY(x_exp < FPBits::EXP_BIAS - 26)) {
39 // When |x| < 2^-26, the relative error of the approximation asin(x) ~ x
40 // is:
41 // |asin(x) - x| / |asin(x)| < |x^3| / (6|x|)
42 // = x^2 / 6
43 // < 2^-54
44 // < epsilon(1)/2.
45 // So the correctly rounded values of asin(x) are:
46 // = x + sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,
47 // or (rounding mode = FE_UPWARD and x is
48 // negative),
49 // = x otherwise.
50 // To simplify the rounding decision and make it more efficient, we use
51 // fma(x, 2^-54, x) instead.
52 // Note: to use the formula x + 2^-54*x to decide the correct rounding, we
53 // do need fma(x, 2^-54, x) to prevent underflow caused by 2^-54*x when
54 // |x| < 2^-1022. For targets without FMA instructions, when x is close to
55 // denormal range, we normalize x,
56#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS)
57 return x;
58#elif defined(LIBC_TARGET_CPU_HAS_FMA_DOUBLE)
59 return fputil::multiply_add(x, 0x1.0p-54, x);
60#else
61 if (xbits.abs().uintval() == 0)
62 return x;
63 // Get sign(x) * min_normal.
64 FPBits eps_bits = FPBits::min_normal();
65 eps_bits.set_sign(xbits.sign());
66 double eps = eps_bits.get_val();
67 double normalize_const = (x_exp == 0) ? eps : 0.0;
68 double scaled_normal =
69 fputil::multiply_add(x: x + normalize_const, y: 0x1.0p54, z: eps);
70 return fputil::multiply_add(x: scaled_normal, y: 0x1.0p-54, z: -normalize_const);
71#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
72 }
73
74#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
75 return x * asin_eval(x * x);
76#else
77 using DFloat128 = fputil::DyadicFloat<128>;
78 using DoubleDouble = fputil::DoubleDouble;
79
80 unsigned idx = 0;
81 DoubleDouble x_sq = fputil::exact_mult(a: x, b: x);
82 double err = xbits.abs().get_val() * 0x1.0p-51;
83 // Polynomial approximation:
84 // p ~ asin(x)/x
85
86 DoubleDouble p = asin_eval(u: x_sq, idx, err);
87 // asin(x) ~ x * (ASIN_COEFFS[idx][0] + p)
88 DoubleDouble r0 = fputil::exact_mult(a: x, b: p.hi);
89 double r_lo = fputil::multiply_add(x, y: p.lo, z: r0.lo);
90
91 // Ziv's accuracy test.
92
93 double r_upper = r0.hi + (r_lo + err);
94 double r_lower = r0.hi + (r_lo - err);
95
96 if (LIBC_LIKELY(r_upper == r_lower))
97 return r_upper;
98
99 // Ziv's accuracy test failed, perform 128-bit calculation.
100
101 // Recalculate mod 1/64.
102 idx = static_cast<unsigned>(fputil::nearest_integer(x: x_sq.hi * 0x1.0p6));
103
104 // Get x^2 - idx/64 exactly. When FMA is available, double-double
105 // multiplication will be correct for all rounding modes. Otherwise we use
106 // DFloat128 directly.
107 DFloat128 x_f128(x);
108
109#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
110 // u = x^2 - idx/64
111 DFloat128 u_hi(
112 fputil::multiply_add(static_cast<double>(idx), -0x1.0p-6, x_sq.hi));
113 DFloat128 u = fputil::quick_add(u_hi, DFloat128(x_sq.lo));
114#else
115 DFloat128 x_sq_f128 = fputil::quick_mul(a: x_f128, b: x_f128);
116 DFloat128 u = fputil::quick_add(
117 a: x_sq_f128, b: DFloat128(static_cast<double>(idx) * (-0x1.0p-6)));
118#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
119
120 DFloat128 p_f128 = asin_eval(u, idx);
121 DFloat128 r = fputil::quick_mul(a: x_f128, b: p_f128);
122
123 return static_cast<double>(r);
124#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
125 }
126 // |x| >= 0.5
127
128 double x_abs = xbits.abs().get_val();
129
130 // Maintaining the sign:
131 constexpr double SIGN[2] = {1.0, -1.0};
132 double x_sign = SIGN[xbits.is_neg()];
133
134 // |x| >= 1
135 if (LIBC_UNLIKELY(x_exp >= FPBits::EXP_BIAS)) {
136 // x = +-1, asin(x) = +- pi/2
137 if (x_abs == 1.0) {
138 // return +- pi/2
139 return fputil::multiply_add(x: x_sign, y: PI_OVER_TWO.hi,
140 z: x_sign * PI_OVER_TWO.lo);
141 }
142 // |x| > 1, return NaN.
143 if (xbits.is_quiet_nan())
144 return x;
145
146 // Set domain error for non-NaN input.
147 if (!xbits.is_nan())
148 fputil::set_errno_if_required(EDOM);
149
150 fputil::raise_except_if_required(FE_INVALID);
151 return FPBits::quiet_nan().get_val();
152 }
153
154 // When |x| >= 0.5, we perform range reduction as follow:
155 //
156 // Assume further that 0.5 <= x < 1, and let:
157 // y = asin(x)
158 // We will use the double angle formula:
159 // cos(2y) = 1 - 2 sin^2(y)
160 // and the complement angle identity:
161 // x = sin(y) = cos(pi/2 - y)
162 // = 1 - 2 sin^2 (pi/4 - y/2)
163 // So:
164 // sin(pi/4 - y/2) = sqrt( (1 - x)/2 )
165 // And hence:
166 // pi/4 - y/2 = asin( sqrt( (1 - x)/2 ) )
167 // Equivalently:
168 // asin(x) = y = pi/2 - 2 * asin( sqrt( (1 - x)/2 ) )
169 // Let u = (1 - x)/2, then:
170 // asin(x) = pi/2 - 2 * asin( sqrt(u) )
171 // Moreover, since 0.5 <= x < 1:
172 // 0 < u <= 1/4, and 0 < sqrt(u) <= 0.5,
173 // And hence we can reuse the same polynomial approximation of asin(x) when
174 // |x| <= 0.5:
175 // asin(x) ~ pi/2 - 2 * sqrt(u) * P(u),
176
177 // u = (1 - |x|)/2
178 double u = fputil::multiply_add(x: x_abs, y: -0.5, z: 0.5);
179 // v_hi + v_lo ~ sqrt(u).
180 // Let:
181 // h = u - v_hi^2 = (sqrt(u) - v_hi) * (sqrt(u) + v_hi)
182 // Then:
183 // sqrt(u) = v_hi + h / (sqrt(u) + v_hi)
184 // ~ v_hi + h / (2 * v_hi)
185 // So we can use:
186 // v_lo = h / (2 * v_hi).
187 // Then,
188 // asin(x) ~ pi/2 - 2*(v_hi + v_lo) * P(u)
189 double v_hi = fputil::sqrt<double>(x: u);
190
191#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
192 double p = asin_eval(u);
193 double r = x_sign * fputil::multiply_add(-2.0 * v_hi, p, PI_OVER_TWO.hi);
194 return r;
195#else
196
197#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
198 double h = fputil::multiply_add(v_hi, -v_hi, u);
199#else
200 DoubleDouble v_hi_sq = fputil::exact_mult(a: v_hi, b: v_hi);
201 double h = (u - v_hi_sq.hi) - v_hi_sq.lo;
202#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
203
204 // Scale v_lo and v_hi by 2 from the formula:
205 // vh = v_hi * 2
206 // vl = 2*v_lo = h / v_hi.
207 double vh = v_hi * 2.0;
208 double vl = h / v_hi;
209
210 // Polynomial approximation:
211 // p ~ asin(sqrt(u))/sqrt(u)
212 unsigned idx = 0;
213 double err = vh * 0x1.0p-51;
214
215 DoubleDouble p = asin_eval(u: DoubleDouble{.lo: 0.0, .hi: u}, idx, err);
216
217 // Perform computations in double-double arithmetic:
218 // asin(x) = pi/2 - (v_hi + v_lo) * (ASIN_COEFFS[idx][0] + p)
219 DoubleDouble r0 = fputil::quick_mult(a: DoubleDouble{.lo: vl, .hi: vh}, b: p);
220 DoubleDouble r = fputil::exact_add(a: PI_OVER_TWO.hi, b: -r0.hi);
221
222 double r_lo = PI_OVER_TWO.lo - r0.lo + r.lo;
223
224 // Ziv's accuracy test.
225
226#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
227 double r_upper = fputil::multiply_add(
228 r.hi, x_sign, fputil::multiply_add(r_lo, x_sign, err));
229 double r_lower = fputil::multiply_add(
230 r.hi, x_sign, fputil::multiply_add(r_lo, x_sign, -err));
231#else
232 r_lo *= x_sign;
233 r.hi *= x_sign;
234 double r_upper = r.hi + (r_lo + err);
235 double r_lower = r.hi + (r_lo - err);
236#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
237
238 if (LIBC_LIKELY(r_upper == r_lower))
239 return r_upper;
240
241 // Ziv's accuracy test failed, we redo the computations in DFloat128.
242 // Recalculate mod 1/64.
243 idx = static_cast<unsigned>(fputil::nearest_integer(x: u * 0x1.0p6));
244
245 // After the first step of Newton-Raphson approximating v = sqrt(u), we have
246 // that:
247 // sqrt(u) = v_hi + h / (sqrt(u) + v_hi)
248 // v_lo = h / (2 * v_hi)
249 // With error:
250 // sqrt(u) - (v_hi + v_lo) = h * ( 1/(sqrt(u) + v_hi) - 1/(2*v_hi) )
251 // = -h^2 / (2*v * (sqrt(u) + v)^2).
252 // Since:
253 // (sqrt(u) + v_hi)^2 ~ (2sqrt(u))^2 = 4u,
254 // we can add another correction term to (v_hi + v_lo) that is:
255 // v_ll = -h^2 / (2*v_hi * 4u)
256 // = -v_lo * (h / 4u)
257 // = -vl * (h / 8u),
258 // making the errors:
259 // sqrt(u) - (v_hi + v_lo + v_ll) = O(h^3)
260 // well beyond 128-bit precision needed.
261
262 // Get the rounding error of vl = 2 * v_lo ~ h / vh
263 // Get full product of vh * vl
264#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
265 double vl_lo = fputil::multiply_add(-v_hi, vl, h) / v_hi;
266#else
267 DoubleDouble vh_vl = fputil::exact_mult(a: v_hi, b: vl);
268 double vl_lo = ((h - vh_vl.hi) - vh_vl.lo) / v_hi;
269#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
270 // vll = 2*v_ll = -vl * (h / (4u)).
271 double t = h * (-0.25) / u;
272 double vll = fputil::multiply_add(x: vl, y: t, z: vl_lo);
273 // m_v = -(v_hi + v_lo + v_ll).
274 DFloat128 m_v = fputil::quick_add(
275 a: DFloat128(vh), b: fputil::quick_add(a: DFloat128(vl), b: DFloat128(vll)));
276 m_v.sign = Sign::NEG;
277
278 // Perform computations in DFloat128:
279 // asin(x) = pi/2 - (v_hi + v_lo + vll) * P(u).
280 DFloat128 y_f128(
281 fputil::multiply_add(x: static_cast<double>(idx), y: -0x1.0p-6, z: u));
282
283 DFloat128 p_f128 = asin_eval(u: y_f128, idx);
284 DFloat128 r0_f128 = fputil::quick_mul(a: m_v, b: p_f128);
285 DFloat128 r_f128 = fputil::quick_add(a: PI_OVER_TWO_F128, b: r0_f128);
286
287 if (xbits.is_neg())
288 r_f128.sign = Sign::NEG;
289
290 return static_cast<double>(r_f128);
291#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
292}
293
294} // namespace math
295
296} // namespace LIBC_NAMESPACE_DECL
297
298#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ASIN_H
299